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 // Copyright ©2016 The Gonum Authors. All rights reserved. // Use of this source code is governed by a BSD-style // license that can be found in the LICENSE file. package gonum import ( "gonum.org/v1/gonum/blas" "gonum.org/v1/gonum/blas/blas64" ) // Dlahr2 reduces the first nb columns of a real general n×(n-k+1) matrix A so // that elements below the k-th subdiagonal are zero. The reduction is performed // by an orthogonal similarity transformation Qᵀ * A * Q. Dlahr2 returns the // matrices V and T which determine Q as a block reflector I - V*T*Vᵀ, and // also the matrix Y = A * V * T. // // The matrix Q is represented as a product of nb elementary reflectors // Q = H_0 * H_1 * ... * H_{nb-1}. // Each H_i has the form // H_i = I - tau[i] * v * vᵀ, // where v is a real vector with v[0:i+k-1] = 0 and v[i+k-1] = 1. v[i+k:n] is // stored on exit in A[i+k+1:n,i]. // // The elements of the vectors v together form the (n-k+1)×nb matrix // V which is needed, with T and Y, to apply the transformation to the // unreduced part of the matrix, using an update of the form // A = (I - V*T*Vᵀ) * (A - Y*Vᵀ). // // On entry, a contains the n×(n-k+1) general matrix A. On return, the elements // on and above the k-th subdiagonal in the first nb columns are overwritten // with the corresponding elements of the reduced matrix; the elements below the // k-th subdiagonal, with the slice tau, represent the matrix Q as a product of // elementary reflectors. The other columns of A are unchanged. // // The contents of A on exit are illustrated by the following example // with n = 7, k = 3 and nb = 2: // [ a a a a a ] // [ a a a a a ] // [ a a a a a ] // [ h h a a a ] // [ v0 h a a a ] // [ v0 v1 a a a ] // [ v0 v1 a a a ] // where a denotes an element of the original matrix A, h denotes a // modified element of the upper Hessenberg matrix H, and vi denotes an // element of the vector defining H_i. // // k is the offset for the reduction. Elements below the k-th subdiagonal in the // first nb columns are reduced to zero. // // nb is the number of columns to be reduced. // // On entry, a represents the n×(n-k+1) matrix A. On return, the elements on and // above the k-th subdiagonal in the first nb columns are overwritten with the // corresponding elements of the reduced matrix. The elements below the k-th // subdiagonal, with the slice tau, represent the matrix Q as a product of // elementary reflectors. The other columns of A are unchanged. // // tau will contain the scalar factors of the elementary reflectors. It must // have length at least nb. // // t and ldt represent the nb×nb upper triangular matrix T, and y and ldy // represent the n×nb matrix Y. // // Dlahr2 is an internal routine. It is exported for testing purposes. func (impl Implementation) Dlahr2(n, k, nb int, a []float64, lda int, tau, t []float64, ldt int, y []float64, ldy int) { switch { case n < 0: panic(nLT0) case k < 0: panic(kLT0) case nb < 0: panic(nbLT0) case nb > n: panic(nbGTN) case lda < max(1, n-k+1): panic(badLdA) case ldt < max(1, nb): panic(badLdT) case ldy < max(1, nb): panic(badLdY) } // Quick return if possible. if n < 0 { return } switch { case len(a) < (n-1)*lda+n-k+1: panic(shortA) case len(tau) < nb: panic(shortTau) case len(t) < (nb-1)*ldt+nb: panic(shortT) case len(y) < (n-1)*ldy+nb: panic(shortY) } // Quick return if possible. if n == 1 { return } bi := blas64.Implementation() var ei float64 for i := 0; i < nb; i++ { if i > 0 { // Update A[k:n,i]. // Update i-th column of A - Y * Vᵀ. bi.Dgemv(blas.NoTrans, n-k, i, -1, y[k*ldy:], ldy, a[(k+i-1)*lda:], 1, 1, a[k*lda+i:], lda) // Apply I - V * Tᵀ * Vᵀ to this column (call it b) // from the left, using the last column of T as // workspace. // Let V = [ V1 ] and b = [ b1 ] (first i rows) // [ V2 ] [ b2 ] // where V1 is unit lower triangular. // // w := V1ᵀ * b1. bi.Dcopy(i, a[k*lda+i:], lda, t[nb-1:], ldt) bi.Dtrmv(blas.Lower, blas.Trans, blas.Unit, i, a[k*lda:], lda, t[nb-1:], ldt) // w := w + V2ᵀ * b2. bi.Dgemv(blas.Trans, n-k-i, i, 1, a[(k+i)*lda:], lda, a[(k+i)*lda+i:], lda, 1, t[nb-1:], ldt) // w := Tᵀ * w. bi.Dtrmv(blas.Upper, blas.Trans, blas.NonUnit, i, t, ldt, t[nb-1:], ldt) // b2 := b2 - V2*w. bi.Dgemv(blas.NoTrans, n-k-i, i, -1, a[(k+i)*lda:], lda, t[nb-1:], ldt, 1, a[(k+i)*lda+i:], lda) // b1 := b1 - V1*w. bi.Dtrmv(blas.Lower, blas.NoTrans, blas.Unit, i, a[k*lda:], lda, t[nb-1:], ldt) bi.Daxpy(i, -1, t[nb-1:], ldt, a[k*lda+i:], lda) a[(k+i-1)*lda+i-1] = ei } // Generate the elementary reflector H_i to annihilate // A[k+i+1:n,i]. ei, tau[i] = impl.Dlarfg(n-k-i, a[(k+i)*lda+i], a[min(k+i+1, n-1)*lda+i:], lda) a[(k+i)*lda+i] = 1 // Compute Y[k:n,i]. bi.Dgemv(blas.NoTrans, n-k, n-k-i, 1, a[k*lda+i+1:], lda, a[(k+i)*lda+i:], lda, 0, y[k*ldy+i:], ldy) bi.Dgemv(blas.Trans, n-k-i, i, 1, a[(k+i)*lda:], lda, a[(k+i)*lda+i:], lda, 0, t[i:], ldt) bi.Dgemv(blas.NoTrans, n-k, i, -1, y[k*ldy:], ldy, t[i:], ldt, 1, y[k*ldy+i:], ldy) bi.Dscal(n-k, tau[i], y[k*ldy+i:], ldy) // Compute T[0:i,i]. bi.Dscal(i, -tau[i], t[i:], ldt) bi.Dtrmv(blas.Upper, blas.NoTrans, blas.NonUnit, i, t, ldt, t[i:], ldt) t[i*ldt+i] = tau[i] } a[(k+nb-1)*lda+nb-1] = ei // Compute Y[0:k,0:nb]. impl.Dlacpy(blas.All, k, nb, a[1:], lda, y, ldy) bi.Dtrmm(blas.Right, blas.Lower, blas.NoTrans, blas.Unit, k, nb, 1, a[k*lda:], lda, y, ldy) if n > k+nb { bi.Dgemm(blas.NoTrans, blas.NoTrans, k, nb, n-k-nb, 1, a[1+nb:], lda, a[(k+nb)*lda:], lda, 1, y, ldy) } bi.Dtrmm(blas.Right, blas.Upper, blas.NoTrans, blas.NonUnit, k, nb, 1, t, ldt, y, ldy) }